Quandle Cocycle Invariants for Knots, Knotted Surfaces and Knotted 3-Manifolds

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Quandle Cocycle Invariants for Knots, Knotted
Surfaces and Knotted 3-Manifolds

• Witold Rosicki (Gdansk)

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Definition A quandle is a set X with a
binary operation (a,b)?a?b such that 1)
For any a?X, a?aa. 2) For any a,b ? X, there
is a unique c? X such that c?ba. 3) For any
a,b,c ? X , we have (a?b)?c(a?c)?(b?c). A
rack is a set with a binary operation that
satisfies 2) and 3). A kei or (involutory
quandle) is a quandle with the additional
property (a?b)?ba for any a,b ?
X. Kei- Takasaki 1942, rack- Conway 1959,
quandle Joyce 1978, Matveev 1982
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• Examples
• 1) X0,1,,n-1, i?j 2j-i mod n.
• XG a group, a?b b-nabn.
• Definition
• Let X be a fixed quandle and let K be a
  given diagram
• of an oriented classical link and let R be the
  set of over-arcs
• (bridges). A quandle coloring is a map cR?X such
  that

c(a)a
c(ß)b
for every crossing.
c(?)a?b
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The Reidemeister moves
I
II
III
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The Reidemeister moves preserves the quandle
coloring.
c?b
I
II
c
a
a
a
a
b
b
a?a
a
c
b
a
b
c
a
a?b
III
a?c
b?c
(a?b)?c
(a?c)?(b?c)
b?c
c
c
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Similarly we can define the quandle coloring for
knotted surfaces
Definition Let X be a fixed quandle and let
K be a given diagram of an oriented knotted
surface in R4 with a given regular projection
pR4?R3 . Let D be the closure of the set of
higher points of the double points of the
projection p and let R be the set of regions,
which we obtain removing D fom our surface. A
quandle coloring is a map cR?X such that
a?b
a
b
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Definition Two knotted surfaces in R4 are
equvalent if there exist an ambiet isotopy of R4
maping one onto other. Theorem (Roseman 1998)
Two knotted surfaces are equivalent iff
one of the broken surface diagram can be obtained
from the other by a finite sequence of moves
from the list of the 7 moves, presented below and
ambient isotopy of the diagrams in 3-space.
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Another presentation of Roseman moves
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The Roseman moves preserves the quandle
coloring, so the quandle coloring is an
invariant of an equivalent class.
Similarly we can define the quandle coloring of
knotted 3-manifolds in 5-space. There
exist 12 Roseman moves such that two knotted
3-manifolds in 5-space are equivalent iff there
exist a finite sequence of these moves between
their diagrams. The Roseman moves preserve
the quandle coloring of 3-manifolds in 5-space.
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Homology and Cohomology Theories of
Quandles. (J.S.Carter, D.Jelsovsky, S.Kamada,
L.Langford and M.Saito 1999, 2003) Let
CRn(X) be the free abelian group generated by
n-tuples (x1,,xn) of elements of a quandle X.
Define a homomorphism ?n CRn(X)?CRn-1(X) by
for n2 and ?n 0 for n1. Then CR?(X) CRn
(X),?n is a chain complex.
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Let CDn(X) be the subset of CRn(X)
generated by n-tuples (x1,,xn ) with xi xi1
or some i ?1,,n-1 if n2 otherwise
CDn(X)0. CD(X) is a sub-complex of
CR(X) CQn(X) CRn(X)/CDn(X) with ?n
induced homomorphism. For an abelian group G,
define the chain and cochain complexes
CW(XG) CW(X) G, ?? id
CW(XG) Hom(CW(X),G),
dHom(?,id) where W D,R,Q.
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As usually, ker ?n ZWn (XG) and im ?n1
BWn(XG) HWn(XG) Hn(CW(XG))
ZWn(XG)/BWn(XG) ker dn ZnW(XG) and im
dn-1 BnW(XG) HnW(XG) Hn(CW(XG))
ZnW(XG)/BnW(XG) Example A function
FXX?G for which the equalities F(x,z)F(x?z,y?z
)F(x?y,z)F(x,y) and F(x,x) 0 are
satisfied for all x,y,z ? X is a quandle
2-cocycle F ? Z2Q(XG)
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The quandle cocycle knot invariant
y
x
?(x,y)
x?y
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c?ba
I
II
-?(a,b)
?(a,a)0
c
?(a,b)
a
a
a
a
b
b
a?a
a
c
b
a
b
c
a
III
a?b
?(b,c)
?(a?b,c)
?(a,b)
?(a?c,b?c)
?(a,c)
a?c
?(b,c)
b?c
(a?b)?c
(a?c)?(b?c)
b?c
c
c
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?(a,c)?(b,c)?(a?c,b?c)?(a,b)?(b,c)?(a?b,c) ?
(a,c)?(a?c,b?c)?(a,b)?(a?b,c) Example (from
picture 14) A function FXX?G for which
the equalities F(x,z)F(x?z,y?z)F(x?y,z)F(x,y)
and F(x,x) 0 are satisfied for all x,y,z ? X is
a quandle 2-cocycle F ? Z (XG)
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b
c
a
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c

(a?c)?(b?c)
b
b?c
a
a?c

III
a?b
(a?b)?c
c
b
b?c
a
c
b
a
b
c
a
a?b
III
a?c
b?c
(a?b)?c
(a?c)?(b?c)
c
c
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?(a?c,b?c)
?(b,c)
b
b?c
a
a?c

?(a,c)
III
?(a?b,c)
a?b
(a?b)?c
c
b
b?c
a
?(a,b)
?(b,c)
?(a,c)?(b,c)?(a?c,b?c)?(a,b)?(b,c)?(a?b,c) ?
(a,c)?(a?c,b?c)?(a,b)?(a?b,c)
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Let C is a given coloring of a knotted surface,
then for each triple point we have assigned a
3-cocycle ?.
a
?(a,b,c)
b
c
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We can define a quandle 3-cocycle invariant of
the position of a surface in a 4-space. The
Roseman moves preserve this invariant.
The first sum is taken over all possible
colorings of the given diagram K of the surface
in 4-space and the second sum (product) is taken
over all triple points.
This theory is described in the book of S.Carter,
S.Kamada and M.Saito Surface in 4-Sace.
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For a knotted 3-manifold in 5-space and its
projection we can define similar
or
where the first sum is taken over all possible
colorings of the given diagram K of the
3-manifold in 5-space and the second sum
(product) is taken over all with multiplicity 4
points. F is an invariant of position
if all 12 Roseman moves preserve it.
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Points with the multiplicity 4 appear only in 3
Roseman moves e, f, l . In e two points
t1, t2 with multiplicity 4 and opposite
orientations appear. Therefore e(t1) -e(t2) and
e(t1)F-e(t2)F0. In l F0 because two colors
must be the same. The calculation in f is
essential. We will calculate similarly on a
picture in 3-space, similarly like we
calculated on a line in the case of a classical
knot. We will project the 4-space onto the
horizontal 3-space. The vertical 3-spaces
will represent as planes. The diagonal 3-space
will project onto whole 3-space. The red triangle
will represent the plane of the intersection of
the horizontal and the diagonal 3-spaces.
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x1
x4
x2
x3
x5
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x4
x1
x1?x3
5
x2
x2?x3
x3
x1?x2
4
3
1
x5
2
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1. F(x2,x3,x4,x5)
2. -F(x1?x2, x3,x4,x5)
3. F(x1?x3,x2?x3,x4,x5)
4. F(x1,x2,x3,x5)
5. -F(x1,x2,x3,x4)

The orientation in the points with multiplicity 4
is given by normal vectors (represented by red
arrows) 1,0,0,0, 0,-1,0,0, 0,0,1,0,
1,-1,1,1, 0,0,0,-1 .
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x4
x2
x3
x1
x5
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x4
x1?x3
x1?x4
x2
x2?x3
x3
x1?x2
x1
2
1
3
4
x5
x2?x5
5
x3?x5
x4?x5
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• F(x2,x3,x4,x5)
• -F(x1, x3,x4,x5)
• 3) F(x1,x2, x4,x5)
• 4) F(x1?x4,,x2?x4,x3?x4,x5)
• 5) -F(x1?x5,x2?x5,x3?x5,x4?x5)

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If F is a 4-cocycle then d(F)(x1,x2,x3,x4,x5)
F(?(x1,x2,x3,x4,x5 )) F(x1,x3,x4,x5) -
F(x1?x2,x3,x4,x5) - F(x1,x2,x4,x5)
F(x1?x3,x2?x3,x4,x5)F(x1,x2,x3,x5)
F(x1?x4,x2?x4,x3?x4,x5) - F (x1,x2,x3,x4)
F(x1?x5,x2?x5,x3?x5,x4?x5) 0
F(x2,x3,x4,x5) -F(x1?x2, x3,x4,x5) F(x1?x3,x2?x
3,x4,x5) F(x1,x2,x3,x5) -F(x1,x2,x3,x4)
F(x2,x3,x4,x5) -F(x1, x3,x4,x5) F(x1,x2,
x4,x5) F(x1?x4,,x2?x4,x3?x4,x5) -F(x1?x5,x2?x5,x3
?x5,x4?x5)

This observation probably will be a part of a
paper which we are going to write with Jozef
Przytycki.